The accuracy and precision in measurement are usually limited by a variety of noise sources. Noise may be caused by sources that are internal or external to a system of interest. For example, instrument noise and environmental noise are external noise. External noise can be reduced by, for example, improving the instrument and isolating the influence of the environment. Internal noise is caused by inherent processes within the system under test. Thermal fluctuations, caused by non-zero operating temperature, and quantum fluctuations, for example, contribute to the internal noise and introduce uncertainty in measured quantities. Such internal noise may be small compared to the external noise in many applications. As the instrument technology and measuring techniques improve, the external noise can sometimes be reduced to a level comparable to or even less than the internal noise. Therefore, in certain applications, the internal noise may become a primary source of noise.
The interaction of an instrument and a sample under measurement can also introduce noise to a measurement. This is called a back action effect. The accuracy or precision of signal detection can be degraded by this back action effect. The measurement problems associated with the internal noise and back action effect can become more significant when the dimension of a system under test or its component parts are microscopic. An example is the damping of the motions of trapped ions in sensitive mass spectrometry measurements by the detection circuit used to detect those motions. See, Lowell S. Brown and Gerald Gabrielse, Geonium Theory: Physics of a Single Electron or Ion in a Penning Trap, Rev. Mod. Phys. 58 (1), 233-311 (1986). This damping may reduce the resolution of the mass spectrum and degrade the precision of the measurement.
The effects of these noise sources in general are present in any system of an arbitrary scale that is sufficiently isolated from its environment so that other noise sources are less important. For devices of a molecular size, the internal nose and the back action effect may be more significant than observation and operation in devices at larger size scales. Noise can be suppressed in a measurement by correlation techniques. A first-order correlation function, for example, is an average of a detected signal such as an electrical current. Since noise is usually random, the first-order correlation function can be used to reduce the effect of the noise. If the signal has a non-zero average, the signal-to-noise ratio can be improved by averaging repetitive measurements of a first-order correlation.
A second-order or higher-order correlation function is an average of a multiplication of two or more measurements of a detected signal. See, e.g., Weissbluth, Photon-Atom Interactions, Academic Press, pp. 276-286, San Diego, Calif. (1989). Such a correlation function may be desirable when characteristic fluctuations in the signal as a function of, for example, time or spatial position provide useful information about the system. Given two measurements of a signal S.sub.b and S.sub.a made at different "positions" q in phase space and/or different times, the second-order correlation function is EQU &lt;S.sub.2 &gt;=&lt;S.sub.b (q.sub.b,t.sub.b)S.sub.a (q.sub.a,t.sub.a)&gt;,
where the bracket indicates an average over repetitions of the measurements. Such a second-order correlation function is of particular importance when the average of a signal over repetitions of the individual factors (i.e., the first-order correlation function) would not depend on either position or time. Higher-order correlation functions may also be used for measurements.
When the two measurements differ only in the times at which they were made, the function is an autocorrelation function. If they are measurements of the same signed quantity measured at two times, then the Fourier transform with respect to the time difference (t.sub.a -t.sub.b) is the power spectrum (or spectral density function) of this quantity. The connection of these quantities as a Fourier pair is the Wiener-Khinchin theorem. If &lt;S.sub.2 &gt; is independent of the time of the initial measurement, the process is said to be stationary, such as a measurement made with the observed system in a steady-state interaction with the measurement apparatus and its environment. When the position and/or the measured quantity differ between the two measurements, the function is called a cross-correlation function.